Condition numbers of indefinite rank 2 ghost Wishart matrices

We define an indefinite Wishart matrix as a matrix of the form A=WTWΣA=WTWΣ, where Σ is an indefinite diagonal matrix and W is a matrix of independent standard normals. We focus on the case where W   is L×2L×2 which has engineering applications [1] and [2]. We obtain the distribution of the ratio of the eigenvalues of A. This distribution can be “folded” to give the distribution of the condition number (Eq. (14)). We calculate formulas for W   real (β=1)(β=1), complex (β=2)(β=2), quaternionic (β=4)(β=4) or any ghost 0<β<∞0<β<∞. We then corroborate our work by comparing them against numerical experiments.